Rationality Quotes May 2013

katydee

Here's another installment of rationality quotes. The usual rules apply:

Please post all quotes separately, so that they can be upvoted or downvoted separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
Do not quote yourself.
Do not quote from Less Wrong itself, Overcoming Bias, or HPMoR.
No more than 5 quotes per person per monthly thread, please.

Here's another installment of rationality quotes. The usual rules apply:

Please post all quotes separately, so that they can be upvoted or downvoted separately. (If they are strongly related, reply to your own comments. If strongly ordered, then go ahead and post them together.)
Do not quote yourself.
Do not quote from Less Wrong itself, Overcoming Bias, or HPMoR.
No more than 5 quotes per person per monthly thread, please.

This is often a good idea in mathematics. Two concepts that are equivalent in some context may no longer be equivalent once you move to a more general context; for example, familiar equivalent definitions are often no longer equivalent if you start dropping axioms from set theory or logic (e.g. the axiom of choice or excluded middle).

Outside of mathematical logic, some familiar examples include:

compactness vs. sequential compactness—generalizing from metric to topological spaces
product topology vs. box topology—generalizing from finite to infinite product spaces
finite-dimensional vs. finitely generated (and related notions, e.g. finitely cogenerated)—generalizing from vector spaces to modules
pointwise convergence vs. uniform convergence vs. norm-convergence vs. convergence in the weak topology vs....—generalizing from sequences of numbers to sequences of functions

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Rationality Quotes May 2013

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Rationality Quotes May 2013

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